# Properties

 Label 63.4.e.a Level $63$ Weight $4$ Character orbit 63.e Analytic conductor $3.717$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 63.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.71712033036$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -3 + 3 \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -3 + 3 \zeta_{6} ) q^{5} + ( -14 + 21 \zeta_{6} ) q^{7} -21 q^{8} +O(q^{10})$$ $$q + ( -3 + 3 \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -3 + 3 \zeta_{6} ) q^{5} + ( -14 + 21 \zeta_{6} ) q^{7} -21 q^{8} -9 \zeta_{6} q^{10} -15 \zeta_{6} q^{11} -64 q^{13} + ( -21 - 42 \zeta_{6} ) q^{14} + ( 71 - 71 \zeta_{6} ) q^{16} + 84 \zeta_{6} q^{17} + ( 16 - 16 \zeta_{6} ) q^{19} + 3 q^{20} + 45 q^{22} + ( -84 + 84 \zeta_{6} ) q^{23} + 116 \zeta_{6} q^{25} + ( 192 - 192 \zeta_{6} ) q^{26} + ( 21 - 7 \zeta_{6} ) q^{28} + 297 q^{29} + 253 \zeta_{6} q^{31} + 45 \zeta_{6} q^{32} -252 q^{34} + ( -21 - 42 \zeta_{6} ) q^{35} + ( 316 - 316 \zeta_{6} ) q^{37} + 48 \zeta_{6} q^{38} + ( 63 - 63 \zeta_{6} ) q^{40} -360 q^{41} + 26 q^{43} + ( -15 + 15 \zeta_{6} ) q^{44} -252 \zeta_{6} q^{46} + ( -30 + 30 \zeta_{6} ) q^{47} + ( -245 - 147 \zeta_{6} ) q^{49} -348 q^{50} + 64 \zeta_{6} q^{52} + 363 \zeta_{6} q^{53} + 45 q^{55} + ( 294 - 441 \zeta_{6} ) q^{56} + ( -891 + 891 \zeta_{6} ) q^{58} -15 \zeta_{6} q^{59} + ( 118 - 118 \zeta_{6} ) q^{61} -759 q^{62} + 433 q^{64} + ( 192 - 192 \zeta_{6} ) q^{65} + 370 \zeta_{6} q^{67} + ( 84 - 84 \zeta_{6} ) q^{68} + ( 189 - 63 \zeta_{6} ) q^{70} + 342 q^{71} -362 \zeta_{6} q^{73} + 948 \zeta_{6} q^{74} -16 q^{76} + ( 315 - 105 \zeta_{6} ) q^{77} + ( -467 + 467 \zeta_{6} ) q^{79} + 213 \zeta_{6} q^{80} + ( 1080 - 1080 \zeta_{6} ) q^{82} -477 q^{83} -252 q^{85} + ( -78 + 78 \zeta_{6} ) q^{86} + 315 \zeta_{6} q^{88} + ( 906 - 906 \zeta_{6} ) q^{89} + ( 896 - 1344 \zeta_{6} ) q^{91} + 84 q^{92} -90 \zeta_{6} q^{94} + 48 \zeta_{6} q^{95} + 503 q^{97} + ( 1176 - 735 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{2} - q^{4} - 3q^{5} - 7q^{7} - 42q^{8} + O(q^{10})$$ $$2q - 3q^{2} - q^{4} - 3q^{5} - 7q^{7} - 42q^{8} - 9q^{10} - 15q^{11} - 128q^{13} - 84q^{14} + 71q^{16} + 84q^{17} + 16q^{19} + 6q^{20} + 90q^{22} - 84q^{23} + 116q^{25} + 192q^{26} + 35q^{28} + 594q^{29} + 253q^{31} + 45q^{32} - 504q^{34} - 84q^{35} + 316q^{37} + 48q^{38} + 63q^{40} - 720q^{41} + 52q^{43} - 15q^{44} - 252q^{46} - 30q^{47} - 637q^{49} - 696q^{50} + 64q^{52} + 363q^{53} + 90q^{55} + 147q^{56} - 891q^{58} - 15q^{59} + 118q^{61} - 1518q^{62} + 866q^{64} + 192q^{65} + 370q^{67} + 84q^{68} + 315q^{70} + 684q^{71} - 362q^{73} + 948q^{74} - 32q^{76} + 525q^{77} - 467q^{79} + 213q^{80} + 1080q^{82} - 954q^{83} - 504q^{85} - 78q^{86} + 315q^{88} + 906q^{89} + 448q^{91} + 168q^{92} - 90q^{94} + 48q^{95} + 1006q^{97} + 1617q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$$.

 $$n$$ $$10$$ $$29$$ $$\chi(n)$$ $$-1 + \zeta_{6}$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.50000 + 2.59808i 0 −0.500000 0.866025i −1.50000 + 2.59808i 0 −3.50000 + 18.1865i −21.0000 0 −4.50000 7.79423i
46.1 −1.50000 2.59808i 0 −0.500000 + 0.866025i −1.50000 2.59808i 0 −3.50000 18.1865i −21.0000 0 −4.50000 + 7.79423i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.4.e.a 2
3.b odd 2 1 21.4.e.a 2
7.b odd 2 1 441.4.e.c 2
7.c even 3 1 inner 63.4.e.a 2
7.c even 3 1 441.4.a.l 1
7.d odd 6 1 441.4.a.k 1
7.d odd 6 1 441.4.e.c 2
12.b even 2 1 336.4.q.e 2
21.c even 2 1 147.4.e.h 2
21.g even 6 1 147.4.a.a 1
21.g even 6 1 147.4.e.h 2
21.h odd 6 1 21.4.e.a 2
21.h odd 6 1 147.4.a.b 1
84.j odd 6 1 2352.4.a.bd 1
84.n even 6 1 336.4.q.e 2
84.n even 6 1 2352.4.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.e.a 2 3.b odd 2 1
21.4.e.a 2 21.h odd 6 1
63.4.e.a 2 1.a even 1 1 trivial
63.4.e.a 2 7.c even 3 1 inner
147.4.a.a 1 21.g even 6 1
147.4.a.b 1 21.h odd 6 1
147.4.e.h 2 21.c even 2 1
147.4.e.h 2 21.g even 6 1
336.4.q.e 2 12.b even 2 1
336.4.q.e 2 84.n even 6 1
441.4.a.k 1 7.d odd 6 1
441.4.a.l 1 7.c even 3 1
441.4.e.c 2 7.b odd 2 1
441.4.e.c 2 7.d odd 6 1
2352.4.a.i 1 84.n even 6 1
2352.4.a.bd 1 84.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 3 T_{2} + 9$$ acting on $$S_{4}^{\mathrm{new}}(63, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 + 3 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$9 + 3 T + T^{2}$$
$7$ $$343 + 7 T + T^{2}$$
$11$ $$225 + 15 T + T^{2}$$
$13$ $$( 64 + T )^{2}$$
$17$ $$7056 - 84 T + T^{2}$$
$19$ $$256 - 16 T + T^{2}$$
$23$ $$7056 + 84 T + T^{2}$$
$29$ $$( -297 + T )^{2}$$
$31$ $$64009 - 253 T + T^{2}$$
$37$ $$99856 - 316 T + T^{2}$$
$41$ $$( 360 + T )^{2}$$
$43$ $$( -26 + T )^{2}$$
$47$ $$900 + 30 T + T^{2}$$
$53$ $$131769 - 363 T + T^{2}$$
$59$ $$225 + 15 T + T^{2}$$
$61$ $$13924 - 118 T + T^{2}$$
$67$ $$136900 - 370 T + T^{2}$$
$71$ $$( -342 + T )^{2}$$
$73$ $$131044 + 362 T + T^{2}$$
$79$ $$218089 + 467 T + T^{2}$$
$83$ $$( 477 + T )^{2}$$
$89$ $$820836 - 906 T + T^{2}$$
$97$ $$( -503 + T )^{2}$$